Array mesh plane modifier with rotation
I'm not sure where your problem comes from but if you want to do that with the Array modifier, here is the way:


Create your card, at the centre of the scene to make it simpler.
In Edit mode shift the mesh from its origin (on the Y axis in my case).
Back in Object mode, create an empty at the same point as the card origin.
Give your card an Array modifier. Choose Object Offset with the empty as object. Choose the Count value you want.
Rotate the empty on the Z axis. In my case, as I've chosen a count of 36, I have to rotate 10° so that it make a complete circle.
As you don't want the cards to be on the same Z level, rise the empty a bit on the Z axis (use shift while you move to make it more precise).